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The entry monomorphism used to start off saying that a monomorphism is an epimorphism in the opposite category…
I have polished and expanded the text now, trying to make it look more like an actual exposition and explanation. I have also expanded a little the Examples-section, and similarly at epimorphism.
These weird kind of entries date from the early days of the Lab, when none of us saw yet what the Lab would once be. Back then it was fun to proceed this way, now it feels awkward.
I hereby pose a challenge to the Forum community:
I challenge you to each pick one entry on a basic topic (nothing fancy), go to the corresponding Lab entry and give it a gentle introductory Idea-section, make sure that the basic motivating examples are mentioned in the order in which the newbie needs to see them, and that the key facts are stated as nicely discernible propositions, best with proof or at least with some helpful pointer, in short, to make the entry a useful read for those readers who would profit from reading it, especially those who do not know the nPOV yet, but might be guided to learn and appreciate it.
Great idea, Urs.
I agree, good idea. There is so much work to do here.
I find “dual of epimorphism” pretty funny. If there’s an order to these concepts, it’s arguably in the other direction: limit notions should come before colimit notions. What I mean is that is epic iff is monic in for all , and similarly all limits and colimits in categories reduce to limits in . Second, for the classical large categories of structures, epimorphisms are harder to understand than monomorphisms.
While proof-reading monomorphism, I see we find ’monic’ used unannounced. I could just replace them all by ’monomorphism’, but how widely used is ’monic’?
Also epimorphism has ’epic’ used (once) unexplained. And ’epi’. Strangely, both seemed to be used as adjectives, where all ’monic’s are nouns.
David, please add a line explaining the jargon.
I use ’monic’ and ’epic’ all the time as adjectives, and ’mono’ and ’epi’ as nouns. I thought the practice was widespread.
So, Todd, would you have ’monos’ instead here, or is there ellipsis for ’monic arrows’?
Frequently, regular and strong monics coincide.
Yes, it should be ’monos’, and I say this even if I was the one who wrote that.
Ok, I think both entries are consistent now.
In the spirit of the challenge in #1, I did add a line explaining the jargon to the Idea-sections at monomorphism, epimorphism and isomorphism.
I had added something to that effect later, there and at epimorphism:
A morphism in some category is called a monomorphism (sometimes abbrieviated to mono), or described as being monic, if …
But maybe better earlier where you have it.
Oh, I didn’t see that, sorry. But it doesn’t hurt to say it in the Idea-section already.
Entry says
Since injective functions are precisely the monomorphisms in Set (example \ref{MonomorphismsInSet} below) this may be stated as saying that is a monomorphism if for all objects then is a monomorphism.
The final “then” feels awkward to my feeling of English, but I am not going to correct it as many native speakers work around. Even logically we can do the analysis. I mean why implication between the quantifier and the clause. “If” is complemented by the preceeding clause “ is a monomorphism”.
I agree with you, Zoran.
I am doing this intentionally to avoid two mathematical formulas appearing in a sentence in the role of two consecutive words. (Even with a comma separating them, this is awkward). I do seem to recall that another native English speaker once active here used to do this, too. But if you tell me that I must be misremembering, then I’ll believe you.
Ah, I see. I think about this point (about adjacent formulas) a lot too. Let me rearrange. (Done.)
That works well now. There are lots of places in papers written in English (and sometimes with native English speakers as authors!) where a very slight change in wording / word order can make a sentence much easier to read, even to parse for its intended meaning. There are questions of personal preference here, even of ’taste’, but, for example, starting a sentence with ’Then’ rarely works well in my view.
Since you still have the lock… you forgot to change “pushout” to “pullback” in proposition 4.1.
Also, if I haven’t gotten turned around, the characterization of epimorphisms and monomorphisms via yoneda lemma is backwards. I.e. the monomorphism page has the version characterizing epis, and the epimorphism page has the version characterizing monos.
below the statement that monomorphisms are preserved under pullback I added a quick pointer to adhesive category for the statement under pushout
added pointer to:
made the link to “injective-on-objects” work
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